Question

If $f:[0,\frac{\pi }{2}]\to [0,\infty ]$ be a function defined by $y=\sin \left(\frac{x}{2}\right)$, then $f$ is

• A. Injective
• B. Surjective
• C. Bijective
• D. None of these

Answer 1

Answer: (A)

Injective

We have $y=\sin \frac{x}{2}$
and $0\le x\le \frac{\pi }{2}$
$\Rightarrow 0\le \frac{x}{2}\le \frac{\pi }{4}$
$\Rightarrow 0\le \sin \frac{x}{2}$ $\le \frac{1}{\sqrt{2}}$
$\Rightarrow (0,\frac{1}{\sqrt{2}})\subset [0,\infty )$
So, function is not surjective but function is injective as for any $0\le x\le \frac{x}{2},\sin \frac{x}{2}$ gives unique image.